The two Friday events are not quite the same. The first-Friday veneration is the older, liturgical one; the weekly display, added in December 2025, simply gives more people the chance to pray before the relic. Between Fridays the Crown is kept in the reliquary shrine in the axial chapel, at the far east end of the cathedral behind the choir.
The cathedral presents the Crown as the relic of the Passion venerated in Jerusalem from the end of the fourth century and later taken to Constantinople. King St. Louis acquired it in 1238, it reached Paris on August 19, 1239, and he built the Sainte-Chapelle to house it. It has been kept in the treasury of Notre-Dame since 1806. The veneration is directed to Christ who suffered, not to the object.
Good to know before you go: entry to the cathedral is free and booking a time slot is optional, but on a Friday afternoon the line can be long, so reserve a free slot on the cathedral's official website or app, or come for the noon Mass and stay. Lent is different: in 2025 the cathedral held the veneration every Friday of Lent, in the afternoon, and it had not published its plans for Lent 2027 when its site was read. One older page on the cathedral's site still shows that 2025 timetable, so rely on the current notice and look again the week you travel.